You need to compete and do beautiful courses. But not only, your horse has to be well-groomed and elegant. Pay attention to the equipment you use to ride and to your outfit! Just by seeing you, the brand must think you can represent them everywhere!
Lauren is absolutely right. There are two scenarios: the brand notices you in competition and ask you to become his ambassador OR you can contact them to know if they are interested in having a representative for their brand. But most of the time, the brand contacts the rider.
First you have to compete often in quite high level to offer more visibility to the brand that will sponsor you. Then you have to be a pretty good rider (do well in competition, be well ranked amongt the other riders...) and be a little bit "known" by the other riders, the coachs and so on.